An RRT-based Navigation Approach for Mobile
Robots and Automated Vehicles
Luis Garrote, Cristiano Premebida, Marco Silva and Urbano Nunes
Institute of Systems and Robotics
Dep. of Electrical and Computer Engineering - University of Coimbra
Coimbra, Portugal
{garrote, cpremebida, msilva, urbano}@isr.uc.pt
Abstract—Advances in autonomous navigation, safety, and
natural-landmark based localization, are among the key objectives in the development of the next generation of autonomous
vehicles, to be deployed in manufacturing and semi-structured
environments. In this paper, autonomous navigation and collision
detection will be focused, where it is proposed a novel navigation
approach that incorporates a RRT-based dynamic path planning
and a path-following controller. Safety issues are taken into
account in the form of a laser-based object detection and tracking.
Experimental results obtained in a virtual environment provide
evidence that our proposed navigation method is promising for
real-world applications.
I. I NTRODUCTION
Recent advances in automated guided vehicles (AGVs), or
laser guided vehicles (LGVs), are towards using AGVs beyond
industrial environments, such as: hospitals, airports, offices,
and intelligent transportation systems. AGVs and mobile robots
share, in general, common issues regarding localization [1], trajectory planning [2], path-following, obstacle avoidance, local
navigation [3], communication, sensor fusion, among others.
However, some current applications impose new requirements
for real-world cases, namely, dynamic path planning, navigation in cluttered environments, and human safety.
Many studies have targeted different aspects of autonomous
vehicles with non-holonomic constraints, such as: kinematics, dynamics, localization, navigation (planning and pathfollowing), controller design, and obstacle avoidance. In this
paper the problem of safe navigation is under focus. Pathfollowing and path planning under safety constraints (collision
avoidance) are particularly addressed.
Path-following has the goal of minimizing a given lateral
error distance between the vehicle and the defined trajectory
(given a-priori or from a path planning algorithm). In general
terms, this minimization is unbounded w.r.t. a time interval,
while maintaining the pose error bounded. The path-following
problem has been well studied in the mobile robotics community, and many solutions have been proposed and applied
in a wide range of cases. Geometric solutions, including the
control law used in the Stanley autonomous vehicle [4] and
Pure Pursuit [5], were proven to be reliable and suitable
to real world situations. In [2], sliding mode control was
applied to differential and car-like robots, providing robust and
stable control laws to systems under uncertainties and external
disturbances. In [6] and [7], Fuzzy logic controllers were
proposed and deployed successfully in autonomous vehicles.
Strictly related to path-following, path planning is a mature
scientific area with proposed solutions ranging from incremental search (e.g., A*, D* light, IDA*) to probabilistic algorithms
(RRT, PRM). The later, probabilistic algorithms, have proven
to be an efficient solution for path planning problems with
kinodynamic constraints in non-convex higher dimensional
spaces. In particular, Rapidly Exploring Random Trees (RRTs)
[8] are among the most popular probabilistic approaches. RRT
works by iteratively creating an exploration tree, starting by
sampling nodes in unexplored regions of the search space, then
finding suitable connections among nodes, and finally adding
new nodes to the exploration tree. The process stops when
a path is found or when a maximum number of iterations is
reached. Building on the initial RRT framework, many methods
have been proposed [9], [10] and applied successfully in realworld scenarios [11], [12].
The development of reliable systems for detection and
tracking of moving objects (DATMO) is a key research goal
in robotics and computer vision as it would greatly facilitate
practical and real-world deployment of automatic safety systems for robots, AGVs, intelligent vehicles, and so on. Human
detection, crucial in safety systems, saw much progress over
the last decade evidenced by various works including [13],
[14], [15]. Nevertheless, there is much room for improvement
when algorithms are tested on realistic scenes, specially when
solutions should be in compliance with safety requirements.
This work brings contributions in the navigation research
field, where a novel approach incorporating a RRT-based
planning and a path-following controller are presented. A
laser scanner based DATMO system is addressed in section
II. Section III-A details our RRT-based local path planning
approach, including collision detection and obstacle avoidance.
In section III-B our proposed path-following method is introduced. Discussion and experimental results using an in-house
C/C++ based simulator are reported in section IV, before the
paper concludes in section V.
II. DATMO
Assuming a single 2D laser scanner is mounted in the front
part of an instrumented vehicle, this section deals with a
DATMO system which is composed of the following principal
stages: (1) segmentation, (2) modeling, and (3) tracking.
1) Segmentation for obstacle detection: In general terms,
segmentation is here defined as the process of separating
foreground objects from the background in the laser scanner
measurement space. The key step is to detect a breakpoint,
characterized by a discontinuity between two consecutive
laser points, which represents, possibly, an object boundary.
Segmentation is a decisive processing phase since failures
during this stage will strongly affect all the subsequent stages,
with a not easy retrofit correction solution. A tradeoff exists in
the segmentation process, which consists in deciding between
merging or splitting the spatial distributed laser-points. In
describing a segmentation process based on 2D range data,
a given segment Sj is defined by the set of laser-points,
sharing similar spatial properties that respect a given clustering
condition. Therefore, Sj is the set of points conditioned on the
parameters of a given (chosen) segmentation method.
Among several methods that can be used for 2D laser data
segmentation, see [16] for a short review, here we briefly describe the segmentation method introduced in [17] which uses
a stochastic filter. This method, called KF-based Breakpoint
Detector (KFBD), uses the Kalman filter (KF) in conjunction
with a statistical test, assuming a Chi-square validation region,
to detect breakpoints. The stochastic model used to describe
the spatial-dynamic evolution of the range measurements as
well as the transition matrices are described in [17]. Basically,
a breakpoint is detected if the normalized innovation squared
exceeds a threshold T hrχ according to a χ21 distribution table.
The model used in the KFBD method is not restrictive w.r.t.
shape thus, it assumes a constant rate of change between
the range-distance and the angle (also termed constant speed
model).
2) Object modeling: Given the set of segments {S} :
Sj ∈ S obtained by the segmentation stage, at this point the
purpose is to represent each segment Sj by a compact and
proper geometric primitive. Among the possibilities (e.g., linesegments, circle, rectangle, ellipse, quadrics), we chose circle
as the primary primitive to represent a segment. Thus, after
circle extraction using the method presented in [18], a segment
Sj is represented by three parameters (xc , yc , rd): center
and radius. Although circle is a suitable representation for
many categories of objects (e.g., poles, pillars, humans, small
machines), in some cases rectangles can be used instead circles,
specially when vehicle-like obstacles are detected. When time
is under consideration, which is particularly important in the
tracking stage, a segment will be explicitly represented by
Sj (k), otherwise the time-index (k) will be omitted.
3) Object tracking: Under linear discrete-time varying
stochastic assumptions, objects dynamics are assumed to
evolve according to a second-order kinematic model, driven
by white noise. The position of a detected object/obstacle,
assuming that it is defined by the circle center (xc , yc ) extracted from a segment Sj (k), is considered to evolve in time
constrained to piecewise constant white noise acceleration [19]
(constrained to maximum values of acceleration). This model
can be understood, in more general terms, in the sense that
the corresponding second-order derivative of the position is
actually not zero (as in a theoretically noiseless model), but
a zero-mean random process entering into the system in the
form of random input noise.
The model assumptions, considering a sample-time interval
h, are that the object keeps a constant acceleration during h:
the noise processes are uncorrelated from period to period
(piecewise indication). Using the KF and a global nearestneighbor data association strategy, our tracking approach is
posed in terms of a decoupled solution, that is, the motion
along each coordinate is assumed decoupled (independent)
from the other coordinates. More specifically, each coordinate
(x, y) of the 2D-Cartesian space is governed by its own equation, with noises entering into x and y coordinates assumed to
be mutually independent with possibly different variances.
III. AUTONOMOUS NAVIGATION
In this section we introduce our autonomous navigation
approach, which encompass dynamic path planning (based
on the RRT* [9] algorithm), collision detection, and pathfollowing. The functional block-diagram of the navigation
system is shown in Fig. 1, having as inputs the set of tracked
obstacles, the desired path and the vehicle pose (2D position
and orientation).
When dealing with scenarios such as hospitals, airports,
offices or structured industrial environments, and assuming
a prior planned route/mission to follow, a number of nonsystematic disturbances can arise. Static obstacles, people and
objects (e.g., vehicles) in motion require different behaviors
to be taken into account by the navigation system. Using
information from the DATMO in the form of tracked obstacles
(see sec. II), knowing the vehicle pose and the desired path to
follow, the navigation system is basically in charge of local
dynamic path planning and path-following.
In situations of obstructed or infeasible paths, safety measures have to be ensured. In this regard, we defined a set of
rules that constitute the baseline of our safety system for semistructured environments:
• Minimum safe distance: the autonomous vehicle must
keep a minimum distance from the infrastructure and
obstacles in the pathway.
• Bounded lateral error: a small error while following a path
must be guaranteed (< 0.5 m).
• Obstacle avoidance: avoidance maneuvers can only be
taken if there is no risk of collision.
The rules listed above are continuously checked and, if an
unsafe condition exists, the current local path is re-planned
according to the dynamic path planning approach represented
in Fig. 2. If a valid solution is found, the new (re-planed) path
is sent to the path-following controller, otherwise the vehicle
is safely immobilized.
A. Dynamic Path Planning
The dynamic path planning method proposed here is based
on a modified version (summarized in Fig. 3) of the original
Fig. 1. Functional block-diagram of the vehicle’s navigation system with the
dynamic path planning and path-following blocks highlighted in bold.
K
K+1
K+n
K+1
K
A
B
Fig. 2. The automated vehicle is represented by the green-box, while a
detected object is represented by the red-circle. Estimations ahead are shown
in dashed line. In the case of a predicted collision, a new valid path is created.
RRT* presented in [9]. The RRT* is a variant of the RRT
algorithm that has an almost sure convergence to an optimal
solution (asymptotic optimality property). As in the original
RRT* algorithm, each node of the RRT contains the vehicle
pose and the last control command. Nevertheless, we introduced two additional parameters: a time frame element and a
penalization term. The time frame element is added to each
RRT node for collision check (used in routine CollisionFree
in Figs. 3 and 6) where a time estimate is maintained from
the root node (beginning of RRT tree); this element is updated
using the routine UpdateTimeFrame using the vehicle speed
estimate (or speed profile) and the traveled distance between
a given node and its parent node.
The penalization term, updated by routine PenalizeNode,
contains the number of invalid expansions already performed
and is used to assess if a node can be further explored. If the
term exceeds a given threshold (in our experiments, a threshold
of 5 failed explorations was defined) the node is considered to
be unexplorable and assumed to be in a intersection zone. The
intersection zone is here defined by all the nodes marked as
unexplorable.
Furthermore, our modified sampling routine (SampleFree)
uses the penalty term to bias the search towards the free space
(and goal) and away from the intersection zone: pseudocode
is shown in Fig. 4.
An important task in dynamic path planning is the capability to perform collision detection (with static and dynamic
obstacles) and guide the search to a valid and collision-free
path.
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Input - Initial configuration : xinit
Maximum number of vertices in RRT : K
Output - RRT graph : G
G.init(xinit )
for k=1 to K do
xrand ← SampleFree(G)
xnearest ← Nearest(xrand , G = (V, E))
xnew ← Steer(xnearest , xrand )
UpdateTimeFrame(xnearest ,xnew )
if CollisionFree(xnearest ,xnew ) then
radius ← min {γRRT ∗ (log(card(V ))/card(V ))1/d , η}
xnear ←SNear(G = (V, E),xnew ,radius)
V ←V
{xnew }
xmin ← xnearest
cmin = Cost(xnearest )+Cost(xnearest ,xnew )
for all x ∈ xnear do
V
if CollisionFree(x,xnew )
Cost(x) + Cost(x,xnew ) < cmin
then
xmin ← x
cmin ← Cost(x)+Cost(x,xnew )
UpdateTimeFrame(x
min ,xnew )
S
E =E
{(xmin ,xnew )}
end if
end for
for all x ∈ xnear do
V
if (CollisionFree(x,xnew )
Cost(xnew ) + Cost(x,xnew ) <
Cost(x) then
xnewP arent ← Parent(x)
UpdateTimeFrame(xnew ,x) S
E ← (E \{xnewP arent ,x})
{(xnew ,x)}
end if
end for
else
PenalizeNode(xnearest )
end if
end for
Fig. 3. RRT* algorithm adapted from [9] - in this paper the routines
PenalizeNode and UpdateTimeFrame were introduced while the routines
CollisionFree and SampleFree from RRT* were modified.
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Input - RRT tree : G
Output - RRT node xrand
xrand ← ∅
while Penalization(Nearest(xrand )) > M axP enalization do
xrand ← RandomNode(G.Goal)
end while
Fig. 4. SampleFree altered routine uses the nearest node (within a search
threshold) to assess if a new node is a RRT node candidate.
Focusing on collision problems like the one described
in [20], in this paper the proposed approach for collision
check (summarized by routine CollisionFree in Fig. 6), was
implemented in such a way that for each node expansion
on the RRT algorithm the dynamic behavior of obstacles is
taken into account. Moreover, each RRT-node contains motion
information (last control command and time frame), meaning
that from a node to another, the position of the vehicle can be
predicted with a bearable uncertainty margin. In particular, we
assumed that a given detected object/obstacle is geometrically
represented by a circle and by its estimated velocity (as previously detailed in sect. II-1). On the other hand, as illustrated
in Fig. 5, the vehicle is defined by a 2D rectangle where the
boundaries (vehicle width and length) are extended by a factor
that depends on the safe emergency stoppage ∆s criterion (see
and a moving obstacle. Considering an approaching obstacle
(dashed circles), and the vehicle predicted motion (dashed
rectangles at t0 and t1 ): the three line segments (L1 , L2 , L3
) and the predicted vehicle boundaries (generated during the
Projection and ComputeVehicleBoundingBox routines (Fig.
6)) are then tested for intersection. This procedure is used at
each RRT iteration, and without a high computational burden
the dynamic path planning can produce feasible solutions in
dynamic scenarios.
B. Path-following Using Motion Primitives
Fig. 5. CollisionFree - When testing if a new edge of the RRT is collision
free, projections from the parent-node (at t0 ) to the expanded node ahead (at
t1 ) are checked.
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Input - Parent Node : xnearest
New Node : xnew
Output - Collision State : state
state ← f alse
box ← ComputeVehicleBoundingBox(xnearest ,xnew ) [eq. 4]
for all Detected Obstacles i (DATMO) do
if i in motion then
L1 ,L2 ,L3 ← Projection(i,Time(xnearest ),Time(xnew )) [Fig. 5]
state ← HasIntersection(box,L1 ,L2 ,L3 )
else
state ← HasIntersection(xnearest ,xnew ,i)
end if
if state is true then
return
end if
end for
Fig. 6. Our CollisionFree routine checks if two nodes (xnearest at t0 and
xnew at t1 in Fig. 5) collide with any of the detected obstacles.
eq. 4). Under ideal conditions, the autonomous vehicle motion
is described by:
vk+1 = vk + ak T
(1)
where v is the vehicle velocity, a is the vehicle acceleration,
T is the time interval (T = Tk+1 − Tk ), with (1) being
constrained by an upper velocity (vmax ): kvk + ak T k ≤ vmax .
The vehicle incremental displacement is then given by:
1
∆ = vk T + ak T 2
(2)
2
If a collision is imminent or an emergency occurs, the
vehicle is forced to stop. Under deterministic assumptions, the
vehicle stops completely when vk+1 = 0, thus:
0 = vk + a k T
(3)
Replacing (3) in (2), the safety distance is given by
This section presents our novel framework for pathfollowing using sampled motion primitives based on the vehicle kinematic model. In this paper, we consider the kinematic
model of a car-like vehicle as detailed in [21], with estimated
position and orientation given by:

 xk+1 = xk + cos θk v
yk+1 = yk + sin θk v
(5)
Xk+1 (u) =

θk+1 = θk + tanLϕk v
where (x, y) designates the 2D position of vehicle’s rear axle
center, θ is the orientation angle w.r.t. the x axis, L is the
distance between axles, ϕ gives the steering angle and v is
the vehicle speed. This model is subject to constraints kϕk ≤
ϕmax and vmin ≤ v ≤ vmax .
The trajectory of an autonomous vehicle can be described
in terms of a set of motion primitives. These primitive motions
(shown in Fig. 7) result from the control commands applied
to the vehicle and, in the sense of path-following, the best
control command is the one that minimizes the error between
the vehicle estimated position and the local desired path.
The path-following approach proposed in this work is described in terms of two sets, M (u) = {X1 , X2 , ...Xn } and
P = {p1 , p2 , ...pg } with respective set sizes n and g. M (u) is
a primitive motion, with length ln (in meters), that corresponds
to the dynamic evolution of the vehicle (eq. 5) according to
the control input (u = {v, ϕ}). P is the desired path, within a
preview window, corresponding to the set of points from the
control point (the center of the rear axle) projected on the road
with length lg (in meters).
Assuming the set U comprises all control input configurations, and the set U 0 is a sampled subset of U containing all
plausible control input configurations, then our path-following
control law is given by the control input uc ∈ U 0 that
minimizes the error between all projected motion primitives
and the desired path:
uc = K1 min0 {h(M (u), P )}
uU
∆s =
vk2
2ak
(4)
Incorporating a safety distance into the planner, as presented
above, makes the collision detection performance more feasible. From a parent node (at t0 ) to a new node (at t1 ), as
exemplified in Fig. 5, the collision detection algorithm checks
for potential intersections between the vehicle bounding box
(6)
where h(M (u), P ) is a measure that represents the similarity
between two sets (e.g., Hausdorff distance [22]) and K1 a proportional gain. The resultant is the vehicle desired command
uc =(vc , ϕc ). Assuming the autonomous vehicle is moving
locally at constant speed and that the allowed speeds are
given as function of the path curvatures, the sampling space
can be restricted to a limited number of acceptable steering
Path
v
(xc,yc )
M(uc)
speed (m/s)
6
4
2
0
300
150
100
250
50
200
0
150
y
y (m)
50
100
50
0
x
Fig. 7. Motion primitives (black lines) obtained from the kinematic model
of a car-like vehicle (Ackerman steering). Desired path is represented by the
solid blue line, while the optimal motion primitive is shown in red.
angles ϕ, where this number decreases proportionally as the
vehicle speed increases. Finally, considering the outputs from
an obstacle detection system, motion primitives that lead to a
risk of collision are discarded: dashed-lines in Fig. 7.
el
)
(7)
v
where eθ is the angular error, el is the lateral error, K1 is a
gain parameter and v is the vehicle speed.
Method 2: with control law of the form [5]:
ϕc = eθ + arctan(K1
2Lel
ϕc = arctan(K1 2 )
(8)
v
where el is the lateral error, L is the distance between axles,
K1 is a gain parameter and v is the vehicle speed.
Method 3: this control law is expressed by [6]:
ϕc = F uzzy System(el , eθ )
(9)
where eθ is the angular error and el is the lateral error. All
angular and lateral errors are computed taking into account
a lookahead distance (La). The lookahead distance is the
distance from the control point (located in the center of rear
axle) to a virtual control point positioned in front of the
vehicle. The lookahead approach and the fuzzy path-following
framework (summarized by the function F uzzy System) are
explained in [6].
To provide a suitable evaluation, four performance measures
are used to compare the path-following controllers: maximum
lateral error (M LE), control effort (CE), mean square error
150
Fig. 8. A path-following simulation scenario with the defined path given in
black, and the speed profile shown in blue.
TABLE I
PATH - FOLLOWING RESULTS CONSIDERING THE SPEED PROFILE AND THE
PATH GIVEN IN F IG . 8.
Method 1 [4]
Method 2 [5]
Method 3 [6]
Proposed
IV. E XPERIMENTS
The evaluation of the path-following method presented in
this paper is based on the vehicle model detailed in [6]. The
autonomous vehicle model was simulated with a maximum
speed of 9 m/s (32.4 Km/h). For performance comparison
purposes, the proposed method is compared with three well
stablished methods:
Method 1: having the control law given by [4]:
x (m)
100
M LE
0.0940
0.2179
0.7655
0.0856
M SE
0.0022
0.0016
0.0825
1.6e-04
CE
0.0338
0.0366
0.1051
0.0351
SV
5.5e-04
0.0015
0.0176
9.7e-04
(M SE) and smoothness variation (SV ). M LE is the maximum measured error between the rear end of the vehicle and
the desired path. M SE is a measure of controller performance
in terms of lateral error, CE is defined by the average of
all absolute control commands demanded by the controller.
Finally, SV represents the variation of the control law, where
small variations means stable transitions. An initial experiment
R
was conducted in a simulated Matlab-based
scenario (given
in Fig. 8) with tight curves and different speed profiles.
Considering this path, the parameters of the evaluated methods
were empirically adjusted:
•
•
•
Method 1 : K1 = 1.6 ; La = 3.0 m
Method 2 : K1 = 0.02 ; La = 2.16 m
Proposed : K1 = 0.95 ; ln = lg = 1.5v
Considering the path shown in Fig. 8, the results obtained by
the path-following methods are summarized in Table I, where
all methods produced smooth trajectories and adapted well to
the defined speed profile. The Method 1 achieved, in general,
a performance as good as the Method 2, but the later had
a higher M LE. On the other hand, Method 3 showed the
highest lateral error because of the controller project (fuzzy
variables and rules were designed for low speeds in the range
0.5 − 2m/s [6]). Finally our proposed controller maintained
stable trajectories with small errors in terms of M LE and
M SE.
In order to evaluate the behavior of the navigation approach,
experiments were carried out in a in-house 3D C/C++ simulation software using the road-like scenario shown in Fig. 9.
A desired path was defined and a set of runs were executed.
Using our RRT-based algorithm (Fig. 3), the trajectories performed by the proposed path-follower, with obstacle free, are
given in Fig. 9. Lastly, further experiments were performed
ACKNOWLEDGMENTS
This work is supported in part by the Portuguese Foundation
for Science and Technology (FCT), under grant PTDC/EEAAUT/113818/2009. Luis Garrote is supported by FCT under
grant SFRH/BD/88459/2012.
R EFERENCES
Fig. 9. An overview of simulation experiments conducted using our 3D
simulation software for the evaluation of the proposed navigation solution.
(a) Static obstacle
(b) Moving obstacle
Fig. 10. Navigation approach behavior, using a RRT-based algorithm and a
novel path-following method, in the presence of a static and a moving obstacle.
considering static and moving obstacles, as shown in Figs.
10a and 10b respectively. Based on simulation results, our
RRT-based navigation method showed evidences of being a
promising approach for real world application.
V. C ONCLUSION AND FUTURE WORK
In this paper we presented a navigation system to be
deployed in automated vehicles for autonomous navigation
in dynamic environments. Solutions to the problems of pathfollowing and local path planning are proposed and simulation
results are reported. Experiments have provided a comparison
of path-following controllers, having our controller achieved
interesting results and evidence of being a promising approach
for real world application. Moreover, the RRT-based dynamic
path-planning approach presented in this work provided evidence, in simulation, to be a potential solution in environments with static and moving obstacles. As future work we
plan to extend our study to include an object classification
process, intended to estimate object’s categories in order to
allow machine decision making according to specific safety
requirements in obstacle avoidance. Moreover, experiments
using a robotic platform, such as the ISRobotCar [23], in a
real-world environment constitute part of our ongoing research.
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